Not exactly a bin cage but when I meshed my ZZ2 I used 6MM mesh just to be on the safe side and it's worked out just fine
Mine is a bar chewer though so the smaller the mesh the better for me, I wouldn't go as big as 13x13 mm mesh. Someone else will have a better idea than me if it's too big (I would think mesh that big might encourage them to try to nibble through, it certainly would with mine!) but I thought my experience might count for something at least
With the water bottle I think a lot of people put it in the yellow cube from P@H if they don't make something to mount it on the wall of the bin but it's up to you if you want to get it through the mesh. I hadn't even considered trying that! I hope you'll upload some pics of the finished project